yes
As per the Patron Saint Index, William of York is a saint but not currently a patron saint.
No, Archangel Gabriel is not a patron saint of school teachers.
There is no patron saint of bullying or against bullying.
Yes he is the patron saint of repented thieves.
There is no patron saint of children's health but the Infant of Prague is the patron saint of good health.
The patron saints of unborn children are Saint Joseph and Saint Gerard Majella.
Saint Nicholas of Myra is the patron saint of children, scholars and penitent thieves as well as a number of other causes. For a complete list of his patronages, click on this link.
== == Saint Nicolas
As per the Patron Saints Index, the patron saints of unborn children are Saint Gerard Majella and Saint Joseph.
Patron saint of Christian apprentices, editors, publishers, school children and young people
There are patron saints of schools and patron saints of artists but no patron saint of art schools.
I see no specific saint for healing of children. However, St. Brigid of Ireland is considered as the patron saint of healers.